Order Hyperbolic Partial Differential Equations Research Articles

A numerical study of moving boundary problem involving dual phase lag model of heat mass transfer during immersion frying

In this paper, we developed a dual phase lag model of heat mass transfer during immersion frying of foods. This model is a moving boundary problem of a couple system of second order hyperbolic partial differential equations. Modified finite element Legendre wavelet Galerkin method developed for its solution. The discretization in space and then application of Legendre wavelet Galerkin method converts our problem into a coupled system of generalized Sylvester equations. In a particular case, the solution of the present model is compared with exact solution and are approximately the same. L∞ error decreases as Legendre wavelet basis functions increase and strip size decreases. Whole analysis is done in dimensionless form. The effect of variability of moving interface on Fourier number, Luikov number, Posnov number, Kossovitch number, Biot number and relaxation times is discussed in detail.

Observer and output feedback control for nonlinear ordinary differential equation coupled to an under‐actuated transport partial differential equation

AbstractIn this article, the under‐actuation problem for a class of first order hyperbolic partial differential equation system coupled to a nonlinear ordinary differential equation is considered. An anticollocated observer design and an appropriate output feedback control is developed based on the backstepping methodology. The stability conditions for the nonlinear coupled system are proposed using the Lyapunov theory and are formulated in terms of the LMI constraints. Further, we have extended the stability results and backstepping control design for the fractional‐order coupled system. The stability conditions and the control law proposed in the work are evaluated considering a numerical simulation.

Dissipative Boundary Control for an Adiabatic Plug Flow Reactor With Mass Recycle

In this contribution the stability properties and regulation of a class of convective systems described by first order hyperbolic partial differential equations with boundary recycle is studied. The system’s setting is consistent with the first and second laws of thermodynamics, allowing to use the entropy functional as a storage function and the internal entropy production as the dissipation similarly to Hamiltonian systems, usually not well defined for systems with mass flows. It is found that the difference of the entropy evaluated at the boundaries is directly proportional to the supply rate, fulfilling the dissipation inequality. Furthermore, the dynamics of the entropy balance allow to define a saturated Proportional-Integral controller with a cascade structure: The inner loop tracks an entropy reference, while the outer loop regulates a process variable. The regulation is achieved with a lumped actuator, using continuous measurements at the boundaries. The controller is applied to an adiabatic plug flow reactor with a recycle of the output stream, a configuration known to be potentially unstable with dissociation reactions. Finally, the controller is tested to track a set point facing several disturbances using the recycle rate as the control variable.

Suppression of Oscillations in Two-Class Traffic by Full-State Feedback

This paper develops a full-state feedback controller that damps out oscillations in traffic density and traffic velocity whose dynamical behavior is governed by the linearized two-class Aw-Rascle (AR) model. Thereby, the traffic is considered to be in the congested regime and subdivided in two classes whereas each class represents vehicles with the same size and driver’s behavior. The macroscopic second-order two-class AR model consists of four first order hyperbolic partial differential equations (PDEs) and introduces a concept of area occupancy to depict the mixed density of two-class vehicles in the traffic. Moreover, the linearized model equations show heterodirectional behavior with both positive and negative characteristic speeds in the congested regime. The control objective is to achieve convergence to a constant equilibrium in finite time. The control input is realized by ramp metering acting at the outlet of the considered track section. The backstepping method is employed to design full-state feedback for the 4 × 4 hyperbolic PDEs. The performance of the full-state feedback controller is verified by simulation.

A Novel Nonlinear Second Order Hyperbolic Partial Differential Equation-Based Image Restoration Algorithm With Directional Diffusion

Recently, variational and partial differential equation (PDE)-based algorithms have become very important for image restoration. In this study, we propose a new second order hyperbolic PDE model based on directional diffusion for image restoration. This hyperbolic PDE restoration model can simply diffuse along the edge's tangential direction in the observed image, thereby removing noise while preserving the image edges and fine details, which avoids the staircase effect in the restored image. An effective numerical scheme is proposed for handling the computation of our approach using the finite difference method. Successful image restoration experiments demonstrated that the proposed second order hyperbolic PDE-based model obtains superior performance compared with other models at preserving edges and it avoids the staircase effect.

Nonlinear control of high duty counter-current heat exchangers using reduced order model

In this study, nonlinear control of high duty counter-current heat exchanger is considered. The dynamics of this system can be captured by first order hyperbolic partial differential equations (PDEs), which are stiff due to the high rate of heat transfer. The potential of multi-time scale dynamics is discussed, and model reduction is performed using singular perturbations to obtain non-stiff PDE models which are valid in each time scale. Three controllers are designed and compared in the simulation case study: input/output linearizing controllers on the basis of original model and reduced order model, and a proportional-integral controller. The nonlinear controller designed using the reduced order model showed a superior performance compared to the others, especially in the presence of large modeling errors and disturbances.

On bivectors and jay-vectors

A combination $\mathbf{a}+\mathrm{i}\mathbf{b}$ where ${\mathrm i}^2=-1$ and $\mathbf{a}, \, \mathbf{b}$ are real vectors is called a bivector. Gibbs developed a theory of bivectors, in which he associated an ellipse with each bivector. He obtained results relating pairs of conjugate semi-diameters and in particular considered the implications of the scalar product of two bivectors being zero. This paper is an attempt to develop a similar formulation for hyperbolas by the use of jay-vectors - a jay-vector is a linear combination $\mathbf{a}+\mathrm{j}\mathbf{b}$ of real vectors $\mathbf{a}$ and $\mathbf{b}$, where ${\mathrm j}^2=+1$ but ${\mathrm j}$ is not a real number, so ${\mathrm j}\neq\pm1$. The implications of the vanishing of the scalar product of two jay-vectors is also considered. We show how to generate a triple of conjugate semi-diameters of an ellipsoid from any orthonormal triad. We also see how to generate in a similar manner a triple of conjugate semi-diameters of a hyperboloid and its conjugate hyperboloid. The role of complex rotations (complex orthogonal matrices) is discussed briefly. Application is made to second order elliptic and hyperbolic partial differential equations. Keywords: Split complex numbers, Hyperbolic numbers, Coquaternions, Conjugate semi-diameters, Hyperboloids and ellipsoids, Complex rotations, PDEs MSC (2010) 35J05, 35L10, 74J05

Modelling and Simulation of a Moving Boundary Problem Arising During Immersion Frying of Foods

A single phase lagging model of heat and moisture transfer has been developed in an infinite slab undergoing immersion frying of foods. This model is a moving boundary problem of a system of second order hyperbolic partial differential equations. The solution obtained by using finite element Legendre wavelet Galerkin scheme. The stability analysis of present model and method are discussed in detailed and whole analysis presented in dimensionless form. The effect of variability of Luikov number, Kossovich number and Biot number on frying process are discussed in detail.

Backstepping boundary observer based-control for hyperbolic PDE in rotary drilling system

It is well known that torsional vibrations in oil well system affect the drilling directions and may be inherent for drilling systems. The drill pipe model is described by second order hyperbolic Partial Differential Equation (PDE) with mixed boundary conditions in which a sliding velocity is considered at the top end. In this paper, we consider the problem of boundary observer design for one-dimensional PDE with the usually neglected damping term. The main purpose is the construction of a control law which stabilizes the damped wave PDE, using only boundary measurements. From the Lyapunov theory, we show an exponentially vibration stability of the partially equipped oil well drilling system. The observer-based control law is found using the backstepping approach for second-order hyperbolic PDE. The numerical simulations confirm the effectiveness of the proposed PDE observer based controller.

New classes of non-convolution integral equations arising from Lie symmetry analysis of hyperbolic PDEs

In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial differential equations. Our analysis leads to new integral equations of non-convolution type, which can be solved by classical methods. We derive solutions of these integral equations, which in turn lead to solutions of the associated Cauchy problems.

Feedback control for a class of second order hyperbolic distributed parameter systems

This paper deals with the problem of state feedback control for a class of {the} distributed parameter systems with {the} disturbance term. {And} the considered distributed parameter systems are composed of {the} second order hyperbolic partial differential equations. Two different classes of restrictions on the disturbance term are given, one is that the disturbance term satisfies the linear growth constraint condition to the state variables of the system, and the other is that the disturbance term obeys the bound constraint under the significance of $L_2 $. Based on {a} variable structure method, the state feedback controllers are obtained by means of constructing appropriate Lyapunov functional. The closed-loop systems are globally asymptotically stable on $W^{1,2}(0,1)\times L_2 (0,1)$ space under the effect of the state feedback control laws. Simulation results illustrate the effectiveness of the proposed method.

Investigation of Fluid-structure Interaction by Explicit Central Finite Difference Methods

Fluid-structure interaction (FSI) occurs when the dynamic water hammer forces; cause vibrations in the pipe wall. FSI in pipe systems due to Poisson and junction coupling has been the center of attention in recent years. It causes fluctuations in pressure heads and vibrations in the pipe wall. The governing equations of this phenomenon include a system of first order hyperbolic partial differential equations (PDEs) in terms of hydraulic and structural quantities. In the present paper, a two-step variant of the Lax-Friedrichs (LxF) method, and a method based on the Nessyahu-Tadmor (NT) are used to simulate FSI in a reservoir-pipe-valve system. The computational results are compared with those of the Method of Characteristics (MOC), Godunov's scheme and also with the exact solution of linear hyperbolic four-equation system to verify the proposed numerical solution. The results reveal that the proposed LxF and NT schemes can predict discontinuous in fluid pressure with an acceptable order of accuracy. The independency of time and space steps allows for setting different spatial grid size with a unique time step, thus increasing the accuracy with respect to the conventional MOC. In these schemes, no Riemann problems were solved and hence field-by-field decompositions were avoided which led to reduced run times compared with Godunov scheme.

Leak Detection, Size Estimation and Localization in Pipe Flows

We design a leak detection system consisting of an adaptive observer based on a set of two coupled one dimensional first order hyperbolic partial differential equations governing the flow dynamics. It is assumed that measurements are only available at the inlet and outlet of the pipe. Convergence properties are provided showing that the leak size estimation problem can be tackled independently from how the leak is distributed along the pipe. This is key to obtaining a simple estimation scheme for the leak location. Simulations are provided to demonstrate the ability of the system to detect, quantify and locate leaks.

A new variable mesh method based on non-polynomial spline in compression approximations for 1D quasilinear hyperbolic equations

In this paper, we present a new three-level implicit method of order two in time and three in space on a non-uniform mesh, based on spline in compression approximation for the numerical solution of 1D quasilinear second order hyperbolic partial differential equations. We also discuss the application of the proposed method to a wave equation with singular coefficients. Stability analysis of a linear scheme and convergence analysis of a general nonlinear scheme are also discussed in this paper. Computational results are given to demonstrate the usefulness of the proposed method.

A New Numerical Method Based on Non-Polynomial Spline in Tension Approximations for 1D Quasilinear Hyperbolic Equations on a Variable Mesh

In this paper, we derive a new three level implicit method of order two in time and three in space, based on spline in tension approximation for the numerical solution of one space dimensional quasi-linear second order hyperbolic partial differential equation on a variable mesh. We also study application of the proposed method to wave equation in polar coordinates. High order approximation at first time level is briefly discussed which is applicable to solve problems both on uniform and non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed method.

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

We study a second order hyperbolic initial‐boundary value partial differential equation (PDE) with memory that results in an integro‐differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 548–563, 2016

NUMERICAL SOLUTION OF WAVE EQUATION USING HAAR WAVELET

Wavelet, being a powerful mathematical tool, has been widely used in image digital processing, quantum field theory and numerical analysis. Wave equation is a special partial differential equation arising in numerous engineering problems. Wave equation is a second order hyperbolic partial differential equation and it describes the phenomenon of wave vibration and wave propagation. Consider the partial differential equation of the form

Lax-Friedrich Scheme for the Numerical Simulation of a Traffic Flow Model Based on a Nonlinear Velocity Density Relation

A fluid dynamic traffic flow model based on a non-linear velocity-density function is considered. The model provides a quasi-linear first order hyperbolic partial differential equation which is appended with initial and boundary data and turns out an initial boundary value problem (IBVP). A first order explicit finite difference scheme of the IBVP known as Lax-Friedrich’s scheme for our model is presented and a well-posedness and stability condition of the scheme is established. The numerical scheme is implemented in order to perform the numerical features of error estimation and rate of convergence. Fundamental diagram, density, velocity and flux profiles are presented.

Age-Structured Population Projection of Bangladesh by Using a Partial Differential Model with Quadratic Polynomial Curve Fitting

In this paper, the age-specific population of Bangladesh based on a linear first order (hyperbolic) partial differential equation which is known as Von-Foerster Equation is studied. Applying quadratic polynomial curve fitting, the total population and population density of Bangladesh are projected for the years 2001 to 2050 based on the explicit upwind finite difference scheme for the age-structured population model based on given data (source: BBS & ICDDR, B) for initial value in the year 2001. For each age-group, the future birth rates and death rates are estimated by using quadratic polynomial curve fitting of the data for the years 2001 to 2012. Quadratic polynomial curve fitting is also used for the boundary value as the (0 - 4) age-group population based on the population size of the age-group for the years 2001 to 2012.

Simulation of Traffic Flow Model with Traffic Controller Boundary

This paper considers a fluid dynamic traffic flow model appended with a closure linear velocity-density relationship which provides a first order hyperbolic partial differential equation (PDE) and is treated as an initial boundary value problem (IBVP). We consider the boundary value in such a way that one side of highway treat like there is a traffic controller at that point. We present the analytic solution of the traffic flow model as a Cauchy problem. A numerical simulation of the traffic flow model (IBVP) is performed based on a finite difference scheme for the model with two sided boundary conditions and a suitable numerical scheme for this is the Lax-Friedrichs scheme. Solution figure from our scheme indicates a desired result that amplitude and frequency of cars density and velocity reduces as time grows. Also at traffic controller point, velocity and density values change as desired manner. In further, we also want to introduce anisotropic behavior of cars(to get more realistic picture) which has not been considered here. Doi: 10.12777/ijse.5.1.25-30 [How to cite this article: Sultana, N., Parvin, M. , Sarker, R., Andallah, L.S. (2013). Simulation of Traffic Flow Model with Traffic Controller Boundary. International Journal of Science and Engineering, 5(1),25-30. Doi: 10.12777/ijse.5.1.25-30]